A library of biorthogonal wavelet transforms originated from polynomial splines

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1 A library of biortogonal wavelet transforms originated from polynomial splines Amir Z. Averbuc a and Valery A. Zeludev a a Scool of Computer Science, Tel Aviv University Tel Aviv 69978, Israel ABSTRACT We present a library of biortogonal wavelet transforms and te related library of biortogonal symmetric waveforms. For te construction we use interpolatory, quasiinterpolatory and smooting splines wit finite masks (local splines). On tis base we designed a set of perfect reconstruction infinite and finite impulse response filter banks wit te linear pase property. Te construction is performed in a lifting manner. Te developed tecnique allows to construct wavelet transforms wit arbitrary prescribed properties suc as number of vanising moments, sape of wavelets, frequency resolution. Moreover, te transforms contain some scalar control parameters wic enable teir flexible tuning in eiter time or frequency domains. Te transforms are implemented in a fast manner. Te transforms based on interpolatory splines are implemented troug recursive filtering.we present encouraging results of applicarion of devised wavelet transforms to image compression.. BIORTHOGONAL WAVELET TRANSFORM Te sequences {a(k)} k=, wic belong to te space l, we will call te discrete-time signals. discrete-time signals we denote by S. Te z transform of a signal {a(k)} S is defined as follows: Te space of a(z) = k= z k a(k). Trougout te paper we assume tat z = e iω. We introduce a family of biortogonal wavelet-type transforms tat operate on te signal x = {x(k)} k=, wic we construct troug lifting steps. First we recall a basic lifting sceme of wavelet transform. Te lifting sceme can be implemented in a primal or dual modes. For brevity we consider only te primal mode... Lifting sceme of wavelet transform.. Decomposition Generally, te primal lifting sceme for decomposition of signals consists of tree steps:. Split.. Predict. 3. Update or lifting. Split - We split te array x into an even and odd sub-arrays: e = {e (k) = x(k)}, d = {d (k) = x(k + )}, k Z. Predict - We use te even array e to predict te odd array d and redefine te array d as te difference between te existing array and te predicted one. To be specific, we apply some filter wit transfer function zu(z) to te sequence e and predict te function d (z ) wic is te z transform of d. Te z transform of te new d array is defined as follows: d u (z ) = d (z ) zu(z)e (z ). (.) From now on te superscript u means an update operation of te array. Obviously, zu(z)e (z ) sould well approximate d (z ). Furter autor information: (Send correspondence to Amir Z. Averbuc ) Amir Z. Averbuc: amir@mat.tau.ac.il Valery A. Zeludev : zel@mat.tau.ac.il

2 Lifting - We update te even array using te new odd array: e u (z ) = e (z ) + β(z)z d u (z ). (.) Generally, te goal of tis step is to eliminate aliasing wic appears wile downsampling te original signal x into e. By doing so we ave tat e is transformed into a low-pass filtered and downsampled replica of x. Furter on we will discuss ow to acieve tis effect by a proper coice of te control filter β, but for now we only require te function β(z) to be real-valued and obey te condition β( z) = β(z), so te product β(z)z is a function of z.... Reconstruction Te reconstruction of te signal x from te arrays e u and d u is implemented in reverse order:. Undo Lifting.. Undo Predict. 3. Unsplit. Undo Lifting - We restore te even array: e (z ) = e u (z ) β(z)z d u (z ). Undo Predict - We restore te odd array: d (z ) = d u (z ) + zu(z)e (z ). Unsplit - Te last step represents te standard restoration of te signal from its even and odd components. In te z domain it looks as: x(z) = e (z ) + z d (z )..3. Filter banks Lifting scemes, tat were presented above, yield efficient algoritms for te implementation of te forward and backward transform of x e u d u. But tese operations can be interpreted as transformations of te signals by a filter bank tat possesses te perfect reconstruction properties. First we define two filter transfer functions Φ l (z) = ( + U(z))/, Define te filter functions g(z) = z Φ (z), Φ (z) = ( U(z))/. β (z) = + β(z)φ (z) = + β(z) z g(z), (z) = Φ l (z) g β (z) = z ( β(z)φ l (z) ) = z ( β(z) (z)). We call β (z) and g(z) te transfer functions of te low-pass and ig-pass primal decomposition filters, respectively. We call (z) and g β (j) te transfer functions of te low-pass and ig-pass primal reconstruction filters, respectively. Tese four filters form a perfect reconstruction filter bank (Ref. 5). Teorem.. Te decomposition and reconstruction formulas can be represented as follows: e u (z ) = (z) x(z) + ( β ) β ( z) x( z) d u (z ) = ( ) g(z) x(z) + g r ( z) x( z) (.3) x(z) = (z)e u (z ) + g β (z) d u (z ). (.) If β( z) = β(z) ten te functions β (j), g(j), (j) and g β (j) satisfy te perfect reconstruction conditions (z) β (z) + g β (z) g(z) = (z) β ( z) + g β (z) g( z) =. Te filter functions are linked to eac oter in te following way: g(z) = z ( z); g β (z) = z β ( z)

3 .. Bases for te signal space Te perfect reconstruction filter banks, tat were constructed above, are associated wit te biortogonal pairs of bases in te space S of discrete-time signals. In section.3 we introduced a family of filters by teir transfer functions (z), g β (z), β (z), g(z). We denote by ϕ (k), ψ β (k), ϕ β (k), ψ (k) te impulse response functions of te corresponding filters, respectively. It means tat, for example r (z) = k Z z k ϕ r, (k) and similarly for te oter functions. Proposition.. Te sifts of te functions ϕ (k), ψ β (k), ϕ β (k) and ψ (k) form a biortogonal pair of bases for te signal space S. Tis means tat any signal x S can be represented as: x(l) = e u (k) ϕ (l k) + d u ψβ(l k). k Z k Z Te coordinates e u (k) and d u (k) can be represented as inner products: e u (k) = x, ϕ β,k, were ϕ β,k(l) = ϕ β(l k) d u (k) = x, ψ k, were ψ k(l) = ψ (l k). Te following biortogonal relations old: ϕ β,k, ϕ l = ψ β,k, ψ l = δ l k, ϕ β,k, ψ β,l = ψ l, ϕ k =, l, k. Te formulated proposition justifies te following definition. Definition.. Te functions ϕ and ψβ, wic belong to te space S, are called te low-frequency and igfrequency syntesis wavelets of te first scale, respectively. Te functions ϕ β and ψ are called te primal lowfrequency and ig-frequency analysis wavelets of te first scale, respectively. Repeated applications of te transform can be acieved in an iterative way. In tis transform we store te array d u and decompose te array e u. Te transformed arrays e u and d u of te second decomposition scale are derived from te even and odd sub-arrays of te array e u by te same lifting steps as tose described above.i As a result we get tat te signal x is transformed into tree subarrays: x d u d u e u. Te reconstruction is performed in te reverse order.. POLYNOMIAL SPLINES We will construct polynomial splines of various kinds using te even subarray of a signal, calculate teir values in te midpoints between nodes and use tese values for prediction of te odd array. In tis section we discuss some properties of suc splines and derive te corresponding filters U... B-splines Te central B spline of te first order on te grid {k} is defined as follows: { M / if x [ /, /] (x) = elsewere. Its Fourier transform is: M (ω) = e iωx M (x)dx = V (ω), V (ω) = sin(ω/) ω/ (.5) Te central B spline of order p is te convolution M p p (x) = M (x) M(x) = π e iωx (V (ω) p dω p. (.6)

4 Note tat te B-spline of order p is supported at te interval ( p/, p/). It is positive witin its support and symmetric around zero. Te nodes of B-splines of even orders are located at points {k} and of te odd orders at points {(k + /)}, k Z. It follows from (.6) tat M p (x) = M p (x), be presented explicitly. Suppose x + = (x + x )/. Ten M p p ( ) p ( (x) = ( ) k x k + p (m )! k We introduce te following sequences k= u p = {M p (k) = M p (k)}, w p = {M p were M p (x) = M p (x). B-splines can ) p +. (.7) ((k + ) ( ) = M p k + )}, k Z (.8) comprised of values of B-splines in te grid points and in te midpoints. Due to te compact support of B-splines, tese sequences are finite. By te reasons to be explained furter on, we will use for our constructions only splines of odd orders p = r. In Tables, we present te sequences for initial values r wic are of practical importance. We need te z transforms of te sequences u p and w p : k u u u Table. Values of te sequences u p. k w 3 w 5 w Table. Values of te sequences w p. u p (z) = k= z k u p (k), w p (z) = k= z k w p (k). Tese functions are Laurent polynomials. Tey are called te Euler-Frobenius polynomials Ref. and were extensively studied in Ref. 3. In particular, te following important facts Ref. 3 were discovered: Proposition.. On te circle z = e iω te Laurent polynomials u p (z) are strictly positive. Teir roots are all simple and negative. Eac root ζ can be paired wit a dual root θ suc tat ζ θ =. Tus if p = r + is odd ten u p (z) can be represented as follows: were < γ n <. We denote u p (z) = r n= γ n ( + γ n z )( + γ n z ), (.9) U p i (z) = z wp (z) u p (z). (.) Proposition.. Te rational functions U p i (z) are real-valued and U p i ( z) = U p i (z). If p = r + is odd ten U p i (z) = (α )r+ ξ r (α) u p (z) were α = z + z and ξ r (α) is a polynomial of degree r., + U p i (z) = ( α )r+ ξ r ( α) u p, (.) (z)

5 .. Interpolatory splines Te sifts of B-splines form a basis in te space S p of splines of order p on te grid k. Namely, any spline Sp Sp as te following representation: S p (x) = q(l) M p (x l). (.) l Denote q = {q(l)} and let q(z ) be te z -transform of q. We introduce also te sequences s p = {S p (k) = Sp (k)} and m p = {S p ((k + /)) = Sp (k + /)} of values of te spline in te grid points and in te midpoints. Let s p (z ) and m p (z ) be corresponding z -transforms. We ave S p (k) = l q(l) M p (k l), Sp ( k + ) = l q(l) M p ( k l + ). (.3) Respectively, s p (z ) = q(z )u(z), m p (z ) = q(z )w(z). From tese formulas we can derive expression for te coefficients of a spline wic interpolates a given sequence e = {e(k)} at grid points: Te z transform of te sequence m p is: S p (k) = e(k), k Z, q(z )u p (z) = e(z ) q(z ) = e(z ) u p (z). (.) m p (z ) = q(z )w p (z) = zu p i (z)e(z ). (.5) Our furter construction exploits te super-convergence property of te interpolatory splines of odd orders (even degrees). Teorem. (Ref. 7). Let a function f L (, ) as p + continuous derivatives and S p Sp interpolates f on te grid {k}. Denote f k = f(k +/)). Ten in te case of odd p = r +, te following asymptotic relation olds S p ((k + /) = f k r+ f (r+) ((k + /)(r + ) b r+() b r+ (/) (r + )! were b s (x) is te Bernoulli polynomial of degree s. + o( r+ f (r+) (x)), (.6) Recall, tat in general te interpolatory spline of order r + approximates te function f wit accuracy of r+. Terefore, we may claim tat {(k + /)} are points of super-convergence of te spline S p. Note, tat te spline of order r + wic interpolates te values of a polynomial of degree r coincides wit tis polynomial. However, te spline of order r + wic interpolates te values of a polynomial of degree r + on te grid {k} restores te values of tis polynomial in te mid-points {(k + /)}. Tis property will result in te vanising moments property of te wavelets to be constructed furter on..3. Quasi-interpolatory splines It is seen from Eqs. (.), (.5) tat, in order to find coefficients of te spline interpolating a signal e and te values of te spline in te midpoints, te signal as to be filtered wit te filters, wose transfer functions are /u p (z) and zu p i (z), respectively. Tese filters ave infinite impulse responses (IIR). However, te property of super-convergence in te midpoints is not exclusive attribute of te interpolatory splines. It is also inerent to te so called local quasi-interpolatory splines of odd orders, wic can be constructed using finite impulse response (FIR) filtering. Definition.. Let a function f as p continuous derivatives and f = {f k = f(k)}, k Z. A spline S p Sp of order p given by (.) is said to be te local quasi-interpolatory spline if te array q of its coefficients is derived by FIR filtering te array of samples f q(z ) = Γ(z )f(z ), (.7)

6 were Γ(z ) is a Laurent polynomial, and te difference f(x) S p (x) = O(f (p) p ).Provided f is a polynomial of degree p, te spline S p (x) f(x). If w p is te sequence defined in (.8) ten te values of te spline S p in te midpoints between grid points mp are produced by te following FIR filtering te array of samples f m p (z ) = zu p (z)f(z ), U p (z) = z Γ(z )w p (z ). (.8) In Ref. 7 explicit formulas for construction of quasi-interpolatory splines are establised as well as te estimations of te differences. In te present work we are interested in te splines of odd orders p = r +. Tere are many FIR filters wic generate quasi-interpolatory splines but only one filter of minimal lengt r + for eac order p = r +. Denote λ(z) = z + z.. Teorem.3. A quasi-interpolatory spline of order p = r + can be produced by filtering (.7) wit filters Γ of lengt no less tan r +. Tere exists a unique filter Γ r m of lengt r +, wic produces te minimal quasi-interpolatory spline (x). Its transfer function is: S r+ Γ r m(z ) = + r βkλ r k (z). (.9) k= Te coefficients β r k can be derived from te generating function ( ) r+ arcsin t/ = ( ) k β r t kt k. (.) If te function f as r + 3 derivatives ten te following asymptotic relations old for te minimal quasiinterpolatory spline S r+ (x) for x = (k + / + τ), τ [, ]: S r+ (x) = f(x) r+ f (r+) (x) b ( ) r+(τ) (r + (r + )! + r+ f (r+) )br+ (τ) (x) βr+ r + O(f (r+3) r+3 ), (r + )! (.) were b s (x) is te Bernoulli polynomial of degree s. We recall tat te values b s = bs () are called te Bernoulli numbers. If s is odd and s > ten b s =. Hence, values of te minimal quasi-interpolatory spline in te midpoints between nodes are te following : S r+ ((k + )) = f((k + )) + r+ f (r+) ((k + ))Ar + O(f (r+3) r+3 ), A r (r + )b r+ = β r (r + )! r+.(.) Tis implies te super-convergence property similar to tat of te interpolatory splines. Te minimal quasiinterpolatory spline of order r + (of degree r +) restores in te midpoints between nodes polynomials of degree r +. Te asymptotic representation (.) allows to enance te approximation accuracy of te minimal spline in te midpoints upgrading te filter Γ r m. Proposition.3. If te coefficients of a spline Šr+ S r+ of order r + are derived in line wit (.7) using te filter Γ r e of lengt r + 3 wit te transfer function Γ r e(z ) = Γ r m(z ) A r λ r+ (z) ten te spline restores polynomials of degree r + 3 in te midpoints between nodes. Tere exists one more way to upgrade te filter Γ r m, wic retains te approximation accuracy of te minimal spline in te midpoints but allows custom design of splines. Proposition. ( Ref. 8). If te coefficients of a spline S r+,ρ S r+ of order r + are derived in line wit (.7) using te filter Γ r ρ of lengt r + 3 wit te transfer function Γ r ρ(z ) = Γ r m(z ) + ρλ r+ (z) ten te spline restores polynomials of degree r + in te midpoints between nodes for any real value ρ. If te parameter ρ is cosen suc tat ρ = ( ) r ρ tan te spline S r+,ρ possesses te smooting property. It means tat it approximates values of te function f in te midpoints between nodes being constructed from te noised samples of te function: f = { f k = f(k) + ε k }, k Z. k=

7 .. Examples First we present a few values of te coefficients β r k : β r = r +, βr = (r + )(r + 7) 576, β3 r = (r + )(r + 6r + 5). 93 For te construction of te extended splines te values of te Bernoulli numbers b s are required. Te initial values are: b = /6, b = /3, b 6 = /.... Quadratic splines Interpolatory spline Denote α = z + z. U i (z) = α z z, U i (α ) (z) = z z, (.3) Minimal spline Te filters are. Γ m(z ) = 8 λ(z), U m(z) = z 3 + 9z + 9z z 3 6, Um(z) = (α ) (z + + z), (.) 6 Extended spline Γ e(z) = Γ m(z ) λ (z), Ue (z) = 3z 5 5z 3 + 5z + 5z 5z 3 + 3z 5, (.5) 56 Ue (z) = (α )3 (3z + 8z z + 3z ), 56 Remark Donoo Ref. presented a sceme were an odd sample is predicted by te value in te central point of te polynomial of an odd degree wic interpolates adjacent even samples. One can observe tat our filter Um coincides wit te filter derived by Donoo s sceme using te cubic interpolatory polynomial. Te filter Ue coincides wit te filter derived using te interpolatory polynomial of fift degree. On te oter and te filter Ui is closely related to te commonly used Butterwort filter Ref.. Namely, in tis case te filter transfer functions Φ,l i (z) = ( + Ui (z))/, Φ, i (z) = ( Ui (z))/. coincide wit magnitude squared te transfer functions of te discrete-time low-pass and ig-pass alf-band Butterwort filters of order, respectively.... Splines of fift order (fourt degree) Interpolatory spline Ui (z) = 6(z3 + z + z + z 3 ) z + 76z z + z, U i (α ) 3 (α ) (z) = z + 76z z. (.6) + z Minimal spline Te filters are Γ m(z ) = 5 7 λ(z) + 5 λ (z), Um(z) = 7(z 7 + z 7 ) + 89(z 5 + z 5 ) 77(z 3 + z 3 ) α 768 Um(z) = (α )3 (7(z + z ) + 8(z 3 + z 3 ) + 76(z + z ) + 366α

8 3. WAVELET TRANSFORMS USING SPLINE FILTERS 3.. Coosing te filters for te lifting step In te previous section we presented a family of te filters U for te predict step wic originate from splines of various types. But, as it is seen from (.), to accomplis te transform we ave to define te filter β. Tere is a remarkable freedom of te coice of tese filters. Te only requirement needed to guarantee te perfect reconstruction property of te transform is β( z) = β(z). In order to make syntesis and analysis filters similar in teir properties, we coose β(z) = Ŭ(z)/, were Ŭ means one of filters U presented above. In particular, Ŭ may coincide wit te filters U wic was used for te prediction. We say tat a wavelet ψ as m vanising moments if te following relations old: k Z ks ψ(k) =, s =,,..., m. Proposition 3.. Suppose tat for te predict and lifting steps filters U(z) and β(z) = Ŭ(z)/ are used, respectively. If U(z) comprises te factor (z + /z) r ten te ig-frequency analysis wavelets ψ ave r vanising moments. If, in addition Ŭ(z) comprises te factor (z + /z)p ten te syntesis wavelet ψβ as q vanising moments, were q = min{p, r}. 3.. Implementation of transforms Once we ave cosen te filter β = Ŭ/, te decomposition procedure is te following (see (.), (.)): d u (z ) = d (z ) zu(z)e (z ), e u (z ) = e (z ) + z Ŭ(z) d u (z ). (3.7) Te transfer function z Ŭ(z)/ differs from zŭ(z)/ only by te factor z, tat is te impulse responses of te corresponding filters are identical up to a sift. Tus bot operations of te decomposition are, in principle, identical. For te reconstruction te same operation are conducted in a reverse order. Terefore, it is sufficient to outline implementation of filtering wit te function zu(z). Tis function depends on z and we denote it as F (z ) = zu(z). Ten te decomposition formulas (3.7) are equivalent to te following d u (z) = d (z) F (z)e (z), e u (z) = e (z) + z F (z) d u (z). (3.8) Equation (3.8) means tat in order to obtain te detail array d u, we must process te even array e by te filter F wit te transfer function F (z) and extract te filtered array from te odd array d. In order to obtain te smooted array e u, we must process te detail array d u by te filter Φ tat as te transfer function Φ(z) = z F (z)/ and add te filtered array to te even array e. But te filter Φ differs from F r / only by one-sample delay and it operates similarly. Implementation of FIR filters originated from local splines is straigtforward and, terefore we only make a few remarks on IIR filters originated from interpolatory splines. Equations (.9) and (.) imply tat, wile te interpolatory spline of order r + is used, te transfer function P (z) F (z) = r n= γ n ( + γ n z)( + γ n z ), (3.9) were P (z) is a Laurent polynomial. It means tat te IIR filter F can be split into a cascade consisting of a FIR filter wit te transfer function P (z), r elementary causal recursive filters, denoted by R(n), and r elementary anti-causal recursive filters, denoted by R(n). On te oter and, te filters can be decomposed into sums of elementary recursive filters. Suc a decomposition also allows parallel implementation of te transform. Te causal and anti-causal filters operate as follows: y = R(n)x y(l) = x(l) + γ n y(l ), y = R(n)x y(l) = x(l) + γ n y(l + ). (3.3)

9 3.3. Examples of recursive filters Now we present IIR filers derived from te interpolatory splines of tird and fift orders. r =. Denote γ = 3.7. Ten Fi (z) = γ + z ( + γ z)( + γ z ). Te filter can be implemented wit te following cascade: x (k) = γ (x(k) + x(k + )), x (k) = x (k) γ x (k ), y(k) = x (k) γ y(k + ). Anoter option stems from te following decomposition of te function F i (z): ( Fi (z) = γ + γ Ten te filter is implemented in parallel mode: + γ + z z + γ z y (k) = x(k) γy (k ), y (k) = x(k + ) γy (k + ), y = γ + γ (y + y ). We note tat elementary filters, wic produce y and y, are operating in opposite directions. r =. Denote ξ = 5 6, γ = , γ = So, we ave te cascade representation: F i (z) = 6γ α ξ + ξz + γ z + ξz + γ z + γ z ). + γ ( + z). z. APPLICATION TO IMAGE COMPRESSION We carried out a series of experiments on compression of various images using a number of wavelet transforms constructed above... Employed wavelet transforms We explored various combinations of filters for te prediction and lifting steps obtaining different transforms. Most of tem are performing satisfactory on compression of images. We display in te following figures wavelets and teir spectra for te tree types of transforms: Ti Te transform were for te prediction and lifting steps te IIR filters U i are used (see (.3) We display in Figure te syntesis and analysis wavelets related to te transform Ti and teir Fourier spectra. One can observe tat te analysis wavelets are regular and, to some extent, are similar to te syntesis ones. Bot te analysis and syntesis ig-frequency wavelets ave vanising moments. Tm Te transform were for te prediction and lifting steps te FIR filters Um are used (see (.) We display in Figure te compactly supported syntesis and analysis wavelets related to te transform Tm and teir Fourier spectra. One can observe tat, unlike te analysis wavelets, te syntesis wavelets are regular and are similar to te syntesis wavelets sown in Figure ones. Bot te analysis and syntesis ig-frequency wavelets ave vanising moments. Ti Te transform were for te prediction and lifting steps te IIR filters U i are used (see (.6) We display in Figure 3 te syntesis and analysis wavelets related to te transform Ti and teir Fourier spectra. One can observe tat te analysis wavelets are regular and are similar to te syntesis ones. Bot te analysis and syntesis ig-frequency wavelets ave 6 vanising moments.

10 Figure. Te left pair bottom to top: Te syntesis wavelets used by te transform Ti : ϕ and ψβ k, k =, 3,, and teir Fourier spectra. Te rigt pair bottom to top: Te analysis wavelets used by te transformti : ϕ β and, k =, 3,, and teir Fourier spectra. ψ k β Figure. Te left pair bottom to top: Te syntesis wavelets used by te transform Tm: ϕ and ψβ k, k =, 3,, and teir Fourier spectra. Te rigt pair bottom to top: Te analysis wavelets used by te transformti : ϕ β and, k =, 3,, and teir Fourier spectra. ψ k β Figure 3. Te left pair bottom to top: Te syntesis wavelets used by te transform Ti : ϕ and ψβ k, k =, 3,, and teir Fourier spectra. Te rigt pair bottom to top: Te analysis wavelets used by te transformti : ϕ β and, k =, 3,, and teir Fourier spectra. ψ k β.. Experiments wit image compression We conducted a wide series of experiments wit compression of multimedia images using te above transforms. In most experiments tey outperform te popular B9/7 biortogonal filter bank. In tis section we present te results after compression and decompression of of four images. Two of tem are commonly used images Lena and Barbara and oter two Car and Fabrics are displayed in Figure. Tese are bit (8bpp) images. Te following experiments were done:

11 Figure. Left: Car, rigt: Fabrics.. We decomposed te image wit te wavelet transform up to 6-t scale using te B9/7 filters and te filters listed in Section... Te transform coefficients were coded using te SPIHT algoritm by Said and Pearlman Ref. 6. Tis algoritm enables to compress data exactly to a prescribed rate. We coded te coefficients wit compression ratios (CR) : (.8 bpp), : (. bpp), :3 (/5 bpp), : (. bpp) and :5 (/5 bpp). 3. Te reconstructed image was compared wit te original image and te peak signal to noise ratio (PSNR) in decibels was computed. ( ) N 55 P SNR = log N k= (x(k) db, x(k)) Lena and Barbara: Te PSNR values of Lena and Barbara images are presented in Table 3. One can observe Lena CR B9/7 Ti Tm Ti Barbara CR B9/7 Ti Tm Ti Table 3. PSNR of te Lena (left table) and Barbara (rigt table) images after te application of SPIHT were te decomposition of te wavelet transforms was into 6 scales. tat for Lena te transform Ti outperforms te B9/7 filter in any compression rate. Te computational complexity of tis transform is sligtly iger tan tat of te B9/7 filter but using parallel mode it can be implemented faster. Te FIR transform Tm, outperform te B9/7 filters in all CR except. Te computational complexity of tis transform is te same as tat of te B9/7 filter but te advantage is tat Tm can be implemented in integers. Computationally more expensive transform Ti in te parallel mode can be implemented faster tan te transform wit B9/7 filter. It is most efficient in te sense of PSNR. Even more it is true for te compression of Barbara. Here Ti remarkably outperforms all oter transforms. Te Tm transform gives way to B9/7. Car and Fabrics : Te PSNR values of Car and Fabrics are presented in Table. On bot images Ti transform outperforms te B9/7 filter for any compression ratio. On Car, wic comprise ig-frequency components,te FIR transform Tm proved to be efficient.

12 Car CR B9/7 Ti Tm Ti Fabrics CR B9/7 Ti Tm Ti Table. PSNR of te Car (left table) and Fabrics (rigt table) images. 5. CONCLUSIONS In tis paper we proposed an efficient metod tat produces new filters for wavelet transforms. Tese wavelet transforms were designed by te usage of polynomial interpolatory and quasi-interpolatory splines. Being applied to te compression of multimedia images, tese filters outperform te traditional biortogonal 9/7 filters wic are frequenty used in wavelet based compression. Te new filters and te biortogonal 9/7 are incorporated into SPIHT in order to measure and compare teir performance wit one well known codec. We presented also wavelet filters wit controlling parameter, wic can be used, in particular for processing noised signals and images. In te future we plan to explore tis opportunity and to ceck te performance of devised filters on video and seismic data. REFERENCES. D. L. Donoo, Interpolating wavelet transform, Preprint 8, Department of Statistics, Stanford University, 99.. I. J. Scoenberg, Cardinal spline interpolation, CBMS,, SIAM, Piladelpia, I. J. Scoenberg, Contribution to te problem of approximation of equidistant data by analytic functions, Quart. Appl. Mat., (96), 5-99, -.. A. V. Oppeneim, R. W. Safer, Discrete-time signal processing, Englewood Cliffs, New York, Prentice Hall, G. Strang, and T. Nguen, Wavelets and filter banks, Wellesley-Cambridge Press, A. Said and W. W. Pearlman, A new, fast and efficient image codec based on set partitioning in ierarcical trees, IEEE Trans. on Circ. and Syst. for Video Tec., 6: 3-5, June V. A. Zeludev Local spline approximation on a uniform grid, U.S.S.R. Comput. Mat. & Mat. Pys., 7 No.5, (987), V.A. Zeludev, Local smooting splines wit a regularizing parameter, Comput. Mat. & Mat Pys., 3, (99), 93-.